Finite Group Databasebeta

← #10595133

Subgroups — #10595133

Order 576.

667 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3805C212yesmaximal, complemented
2D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#940C211yesmaximal, characteristic, complemented
4C3×#938C211yesmaximal, characteristic, complemented
5#67⋊Dic3C212yesmaximal, complemented
6#63970C212yesmaximal, complemented
7#63959C212yesmaximal, complemented
8#63918C211yesmaximal, characteristic, complemented
9#63908C211yesmaximal, characteristic, complemented
10#63898C211yesmaximal, characteristic, complemented
11C4×D3×Dic3C211yesmaximal, characteristic, complemented
12#5716—31nomaximal
13#5596—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×#67C2212yes—
18C2×C3×#67C2212yes—
19C2×C3×D12C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×#627C2212yes—
22C2×#625C2212yes—
23#3805—28no—
24C2×D3×Dic3C2212yes—
25C2×D3×Dic3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27#3757C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C4×C3⋊Dic3C2211yescharacteristic
30C3×#217C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33Dic3⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2211yescharacteristic
37Dic32C2211yescharacteristic
38C2×#237—32no—
39C2×#237—32no—
40C3×#127—31no—
41C3×#121—31no—
42#940D311yescharacteristic, complemented
43#939—32no—
44#938—32no—
45#938—31no—
46#938D311yescharacteristic, complemented
47#936—32no—
48D4×Dic3—31no—
49D4×Dic3—31no—
50D4×Dic3—32no—
51D4×Dic3—31no—
52D4×Dic3—61no—
53#930—31no—
54C2×C4×Dic3—31no—
55#893—31no—
56#888—31no—
57#885—32no—
58#67⋊C4—32no—
59C42×D3—31no—
60#869—31no—
61C23×C32C2312yes—
62C22×C3×D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C32×D4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3—24no—
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2311yescharacteristic
70C3×#67—24no—
71C3×#67—24no—
72C3×#67—24no—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2311yescharacteristic
77C3×D12—22no—
78C3×C4×D3—22no—
79#627—24no—
80#625—24no—
81D3×Dic3—24no—
82D3×Dic3—22no—
83D3×Dic3D414yescomplemented
84#532—91noSylow
85C2×C3×D4—21no—
86C2×C3×D4—32no—
87C2×C3×D4—32no—
88C2×C3×D4—32no—
89C2×C3×D4—31no—
90C2×C3×D4C2×D311yescharacteristic
91C2×C3×D4C2×D311yescharacteristic
92C22×C3×C4—31no—
93C2×#67—32no—
94C2×#67—32no—
95C2×#67C2×D312yes—
96C2×#67C2×D312yes—
97C2×#67C2×D312yes—
98C22×Dic3—31no—
99C22×Dic3—32no—
100C22×Dic3—32no—
101C22×Dic3—31no—
102C22×Dic3—32no—
103C22×Dic3—62no—
104#237—68no—
105#237—68no—
106C2×D12C2×D311yescharacteristic
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3—32no—
110C2×C4×D3C2×D311yescharacteristic
111C2×Dic6—32no—
112C2×Dic6—32no—
113C3×C4⋊C4—31no—
114C3×C22⋊C4—32no—
115C3×C42—31no—
116#217—32no—
117#217—32no—
118#217—32no—
119#217—32no—
120#217—62no—
121#217C2×D312yes—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124C4⋊Dic3—31no—
125C4⋊Dic3—31no—
126C4⋊Dic3—32no—
127C4⋊Dic3—61no—
128C4⋊Dic3C2×D311yescharacteristic
129Dic3⋊C4—32no—
130C4×Dic3—31no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—31no—
136C4×Dic3—61no—
137C4×Dic3C2×D311yescharacteristic
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32C2411yescharacteristic
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—22no—
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3C2×D412yes—
148C32×C4#4112yescomplemented
149C3⋊Dic3—22no—
150C3⋊Dic3—21no—
151C3⋊Dic3C2×D412yes—
152C3×Dic3—22no—
153C3×Dic3—21no—
154C3×Dic3C2×D412yes—
155C3×Dic3#4112yescomplemented
156C2×#41—92no—
157C4⋊Q8—91no—
158#127—31no—
159#124—92no—
160#121—31no—
161#121—91no—
162#121—92no—
163C4×D4—92no—
164C4×D4—91no—
165C4×D4—91no—
166C2×C42—91no—
167C23×C3—22no—
168C23×C3—31no—
169C23×C3—32no—
170C23×C3C22×D312yes—
171C23×C3C22×D312yes—
172C22×D3C22×D312yes—
173C22×D3C22×D311yescharacteristic
174C3×D4—24no—
175C3×D4—22no—
176C3×D4—22no—
177C3×D4—64no—
178C3×D4—64no—
179C3×D4—64no—
180C3×D4—62no—
181C2×C3×C4—21no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—62no—
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67—24no—
190#67—24no—
191#67—24no—
192#67—64no—
193#67—64no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—34no—
197C2×Dic3—31no—
198C2×Dic3—31no—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—64no—
205C2×Dic3—64no—
206C2×Dic3—64no—
207C2×Dic3—62no—
208C2×Dic3—124no—
209C2×Dic3C22×D311yescharacteristic
210C2×Dic3C22×D312yes—
211C2×Dic3C22×D311yescharacteristic
212C2×Dic3C22×D311yescharacteristic
213D12—22no—
214C4×D3—22no—
215C4×D3—34no—
216C4×D3—64no—
217C4×D3—62no—
218Dic6—64no—
219Dic6—64no—
220C2×C32—42no—
221C2×C32C22×D411yescharacteristic
222C2×C32C2×#4111yescharacteristic
223C2×C32C2×#4111yescharacteristic
224C3×D3—22no—
225C3×D3—42no—
226#41—188no—
227C2×Q8—92no—
228C2×D4—32no—
229C2×D4—32no—
230C2×D4—32no—
231C2×D4—31no—
232C2×D4—92no—
233C2×D4D3211yescharacteristic
234C22×C4—31no—
235C22×C4—91no—
236C22×C4—91no—
237C22×C4—92no—
238C22×C4—92no—
239C4⋊C4—31no—
240C4⋊C4—92no—
241C4⋊C4—91no—
242C22⋊C4—32no—
243C22⋊C4—92no—
244C22⋊C4—92no—
245C22⋊C4—92no—
246C42—31no—
247C42—91no—
248C42—91no—
249C42—91no—
250C22×C3—24no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—21no—
258C22×C3—32no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3C23×D311yescharacteristic
267C22×C3C23×D311yescharacteristic
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—22no—
271C2×D3—22no—
272C2×D3—22no—
273C2×D3D3×D412yes—
274C3×C4—22no—
275C3×C4—32no—
276C3×C4—32no—
277C3×C4—62no—
278C3×C4—61no—
279C3×C4#23712yescomplemented
280C3×C4#23712yescomplemented
281Dic3—22no—
282Dic3—21no—
283Dic3—32no—
284Dic3—32no—
285Dic3—62no—
286Dic3—62no—
287Dic3—62no—
288Dic3—61no—
289Dic3—61no—
290Dic3—122no—
291Dic3—121no—
292Dic3D3×D412yes—
293Dic3#23712yescomplemented
294C32#53211yescharacteristic, complemented, Sylow
295C23—31no—
296C23—32no—
297C23C2×D3212yes—
298Q8—184no—
299D4—22no—
300D4—64no—
301D4—64no—
302D4—64no—
303D4—62no—
304D4—184no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—62no—
310C2×C4—94no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—92no—
314C2×C4—182no—
315C2×C4—184no—
316C2×C4—184no—
317C2×C4C2×D3211yescharacteristic
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—44no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3—62no—
325C2×C3—122no—
326C2×C3C2×D3×D411yescharacteristic
327C2×C3C2×D3×D411yescharacteristic
328C2×C3C2×#23711yescharacteristic
329C2×C3C2×#23711yescharacteristic
330C2×C3C2×#23711yescharacteristic
331C2×C3D3×#4111yescharacteristic
332D3—22no—
333D3—42no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—32no—
338C22—61no—
339C22—61no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22C22×D3211yescharacteristic
344C4—32no—
345C4—32no—
346C4—62no—
347C4—61no—
348C4—92no—
349C4—182no—
350C4—181no—
351C4#380512yescomplemented
352C3—21no—
353C3#559611yescharacteristic, complemented
354C3#571611yescharacteristic, complemented
355C2—42no—
356C2—62no—
357C2—122no—
358C2D32×D411yescharacteristic
359C2D3×#23711yescharacteristic
360C2C2×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3805C2semidirect
2. D4×C3⋊Dic3C2semidirect
3. C3×#940C2semidirect
4. C3×#938C2semidirect
5. #67⋊Dic3C2semidirect
6. #63970C2semidirect
7. #63959C2semidirect
8. #63918C2semidirect
9. #63908C2semidirect
10. #63898C2semidirect
11. C4×D3×Dic3C2semidirect
42. #940D3semidirect
46. #938D3semidirect
83. D3×Dic3D4semidirect
148. C32×C4#41semidirect
155. C3×Dic3#41semidirect
279. C3×C4#237semidirect
280. C3×C4#237semidirect
293. Dic3#237semidirect
294. C32#532semidirect
351. C4#3805semidirect
353. C3#5596semidirect
354. C3#5716semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#38052
D4×C3⋊Dic32
C3×#9402
C3×#9382
#67⋊Dic32
#639702
#639592
#639182
#639082
#638982
C4×D3×Dic32
#57163
#55963